Ready for STEP 2 and 3

Show that $(*)$ has three real roots $\alpha$, $\beta$, $\gamma$ with $2$ $< \alpha$ $< 3$, $0$ $< \beta$ $< 1$ and $-1$ $< \gamma$ $< 0$.

Find the cubic equation whose roots are $\frac{1}{\alpha}$, $\frac{1}{\beta}$, $\frac{1}{\gamma}$, and the cubic equation whose roots are $1-\alpha$, $1-\beta$, $1-\gamma$. Deduce that the sets $\left\{\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}\right\}$ and $\{1$${}-\alpha, 1$${}-\beta, 1$${}-\gamma\}$ are equal.

Let $S(x)$ $= 1 - \frac{1}{x}$. Show that $S(\alpha)$, $S(\beta)$ and $S(\gamma)$ are roots of $(*)$, and deduce that $S(\alpha)$ $= \beta$, $S(\beta)$ $= \gamma$ and $S(\gamma)$ $= \alpha$.

Find the value of $\alpha^2\beta$${}+ \beta^2\gamma$${}+ \gamma^2\alpha$.